Quiz — Lycée Pilote Sfax - Identité de Bézout 3.pdf

Question 1 sur 10 20:00

[{"id":37790,"question":"Quelle est la forme générale de l'identité de Bézout pour deux entiers \u003Cem\u003Ea\u003C\/em\u003E et \u003Cem\u003Eb\u003C\/em\u003E ?","option_a":"\u003Cem\u003Ea + b = pgcd(a, b)\u003C\/em\u003E","option_b":"\u003Cem\u003Eau + bv = pgcd(a, b)\u003C\/em\u003E","option_c":"\u003Cem\u003Ea² + b² = pgcd(a, b)\u003C\/em\u003E","option_d":"\u003Cem\u003Ea\/b + b\/a = pgcd(a, b)\u003C\/em\u003E","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'identité de Bézout s'écrit \u003Cem\u003Eau + bv = pgcd(a, b)\u003C\/em\u003E où \u003Cem\u003Eu\u003C\/em\u003E et \u003Cem\u003Ev\u003C\/em\u003E sont des entiers relatifs.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"\u003Cem\u003Ea + b = pgcd(a, b)\u003C\/em\u003E\", \"b\": \"\u003Cem\u003Eau + bv = pgcd(a, b)\u003C\/em\u003E","_debug_options_count":4},{"id":37791,"question":"Si \u003Cem\u003Epgcd(a, b) = 1\u003C\/em\u003E, alors \u003Cem\u003Ea\u003C\/em\u003E et \u003Cem\u003Eb\u003C\/em\u003E sont dits...","option_a":"premiers entre eux","option_b":"premiers","option_c":"pairs","option_d":"impairs","option_e":"","option_f":"","bonne_reponse":"a","explication":"Deux entiers sont premiers entre eux si leur PGCD vaut 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"premiers entre eux\", \"b\": \"premiers\", \"c\": \"pairs\", \"d\": \"impairs","_debug_options_count":4},{"id":37792,"question":"Peut-on résoudre l'équation \u003Cem\u003E3x + 5y = 1\u003C\/em\u003E avec l'identité de Bézout ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, car \u003Cem\u003Epgcd(3, 5) = 1\u003C\/em\u003E, donc l'équation a des solutions entières.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":37793,"question":"Quelle est la solution générale de l'équation \u003Cem\u003E6x + 9y = 3\u003C\/em\u003E ?","option_a":"\u003Cem\u003Ex = 1 + 3k, y = -1 - 2k\u003C\/em\u003E","option_b":"\u003Cem\u003Ex = 2 + 3k, y = -1 - 2k\u003C\/em\u003E","option_c":"\u003Cem\u003Ex = 1 + 2k, y = -1 - 3k\u003C\/em\u003E","option_d":"\u003Cem\u003Ex = 3 + k, y = -2 - k\u003C\/em\u003E","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale est \u003Cem\u003Ex = 1 + 3k, y = -1 - 2k\u003C\/em\u003E pour tout entier \u003Cem\u003Ek\u003C\/em\u003E.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"\u003Cem\u003Ex = 1 + 3k, y = -1 - 2k\u003C\/em\u003E\", \"b\": \"\u003Cem\u003Ex = 2 + 3k, y = -1 -","_debug_options_count":4},{"id":37794,"question":"Si \u003Cem\u003Ea ≡ b [n]\u003C\/em\u003E, alors \u003Cem\u003Ea - b\u003C\/em\u003E est divisible par...","option_a":"\u003Cem\u003Ea\u003C\/em\u003E","option_b":"\u003Cem\u003Eb\u003C\/em\u003E","option_c":"\u003Cem\u003En\u003C\/em\u003E","option_d":"\u003Cem\u003Epgcd(a, n)\u003C\/em\u003E","option_e":"","option_f":"","bonne_reponse":"c","explication":"La congruence \u003Cem\u003Ea ≡ b [n]\u003C\/em\u003E signifie que \u003Cem\u003En\u003C\/em\u003E divise \u003Cem\u003Ea - b\u003C\/em\u003E.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"\u003Cem\u003Ea\u003C\/em\u003E\", \"b\": \"\u003Cem\u003Eb\u003C\/em\u003E\", \"c\": \"\u003Cem\u003En\u003C\/em\u003E\", \"d\": \"\u003Cem\u003Epgcd","_debug_options_count":4},{"id":37795,"question":"L'équation \u003Cem\u003E2x + 4y = 5\u003C\/em\u003E admet-elle des solutions entières ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Non, car \u003Cem\u003Epgcd(2, 4) = 2\u003C\/em\u003E ne divise pas 5. Il n'y a donc pas de solutions entières.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":37796,"question":"Quelle méthode permet de trouver une solution particulière à \u003Cem\u003Eau + bv = pgcd(a, b)\u003C\/em\u003E ?","option_a":"Algorithme de Dijkstra","option_b":"Algorithme d'Euclide étendu","option_c":"Méthode de Cardan","option_d":"Théorème de Pythagore","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'algorithme d'Euclide étendu permet de trouver \u003Cem\u003Eu\u003C\/em\u003E et \u003Cem\u003Ev\u003C\/em\u003E efficacement.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Algorithme de Dijkstra\", \"b\": \"Algorithme d'Euclide étendu\", \"c\"","_debug_options_count":4},{"id":37797,"question":"Si \u003Cem\u003Ea\u003C\/em\u003E et \u003Cem\u003Eb\u003C\/em\u003E sont premiers entre eux, alors \u003Cem\u003Ea\u003C\/em\u003E admet un inverse modulo \u003Cem\u003Eb\u003C\/em\u003E.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, car il existe \u003Cem\u003Eu\u003C\/em\u003E tel que \u003Cem\u003Eau ≡ 1 [b]\u003C\/em\u003E (théorème de Bézout).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":37798,"question":"Quelle est la solution générale de \u003Cem\u003E7x + 3y = 1\u003C\/em\u003E ?","option_a":"\u003Cem\u003Ex = -1 + 3k, y = 2 - 7k\u003C\/em\u003E","option_b":"\u003Cem\u003Ex = 1 + 3k, y = -2 - 7k\u003C\/em\u003E","option_c":"\u003Cem\u003Ex = 2 + 3k, y = -5 - 7k\u003C\/em\u003E","option_d":"\u003Cem\u003Ex = -2 + 3k, y = 5 - 7k\u003C\/em\u003E","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une solution particulière est \u003Cem\u003Ex = -1, y = 2\u003C\/em\u003E, donc la solution générale est \u003Cem\u003Ex = -1 + 3k, y = 2 - 7k\u003C\/em\u003E.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"\u003Cem\u003Ex = -1 + 3k, y = 2 - 7k\u003C\/em\u003E\", \"b\": \"\u003Cem\u003Ex = 1 + 3k, y = -2 -","_debug_options_count":4},{"id":37799,"question":"L'identité de Bézout est-elle valable pour des entiers non nuls ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, l'identité de Bézout s'applique à tout couple d'entiers non nuls.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]